Properties

Label 4-60e2-1.1-c2e2-0-0
Degree $4$
Conductor $3600$
Sign $1$
Analytic cond. $2.67283$
Root an. cond. $1.27862$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s + 4·7-s + 7·9-s + 16·13-s − 68·19-s + 16·21-s − 5·25-s − 8·27-s + 28·31-s + 112·37-s + 64·39-s + 16·43-s − 86·49-s − 272·57-s − 92·61-s + 28·63-s + 64·67-s − 212·73-s − 20·75-s − 44·79-s − 95·81-s + 64·91-s + 112·93-s + 244·97-s − 92·103-s + 172·109-s + 448·111-s + ⋯
L(s)  = 1  + 4/3·3-s + 4/7·7-s + 7/9·9-s + 1.23·13-s − 3.57·19-s + 0.761·21-s − 1/5·25-s − 0.296·27-s + 0.903·31-s + 3.02·37-s + 1.64·39-s + 0.372·43-s − 1.75·49-s − 4.77·57-s − 1.50·61-s + 4/9·63-s + 0.955·67-s − 2.90·73-s − 0.266·75-s − 0.556·79-s − 1.17·81-s + 0.703·91-s + 1.20·93-s + 2.51·97-s − 0.893·103-s + 1.57·109-s + 4.03·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(2.67283\)
Root analytic conductor: \(1.27862\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{60} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3600,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.009681472\)
\(L(\frac12)\) \(\approx\) \(2.009681472\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_2$ \( 1 - 4 T + p^{2} T^{2} \)
5$C_2$ \( 1 + p T^{2} \)
good7$C_2$ \( ( 1 - 2 T + p^{2} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 62 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 - 8 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 398 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 + 34 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 562 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 62 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 - 14 T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 56 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 2642 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 2798 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 3998 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 6782 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 46 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 32 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 7202 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 + 106 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 22 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 802 T^{2} + p^{4} T^{4} \)
89$C_2$ \( ( 1 - 142 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \)
97$C_2$ \( ( 1 - 122 T + p^{2} T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.79968700639781218625433619833, −14.77466625208214471714713709051, −14.15359525255581184575442336732, −13.40238793048262111745947212203, −12.98218510276332119757938288328, −12.73072319139510669475686442289, −11.55079898514535797934876780922, −11.16421606278365258658772828018, −10.53223108714509347242375660529, −9.883757467932475121419756034879, −8.985168264332896447058075911239, −8.631354096503470311805115270511, −8.127246269305760822398265053348, −7.63103121951426138730849172186, −6.29280108209387160444954218493, −6.18085250429145341658873508604, −4.47219991278825650452245220563, −4.14352290029282797770081817173, −2.87606472706765793080168502350, −1.89223480576888286366263656030, 1.89223480576888286366263656030, 2.87606472706765793080168502350, 4.14352290029282797770081817173, 4.47219991278825650452245220563, 6.18085250429145341658873508604, 6.29280108209387160444954218493, 7.63103121951426138730849172186, 8.127246269305760822398265053348, 8.631354096503470311805115270511, 8.985168264332896447058075911239, 9.883757467932475121419756034879, 10.53223108714509347242375660529, 11.16421606278365258658772828018, 11.55079898514535797934876780922, 12.73072319139510669475686442289, 12.98218510276332119757938288328, 13.40238793048262111745947212203, 14.15359525255581184575442336732, 14.77466625208214471714713709051, 14.79968700639781218625433619833

Graph of the $Z$-function along the critical line